压杆稳定性:失稳前后的势能景观

Column Stability: Energy Landscapes Before and After Buckling

  • 摘要: 压杆稳定性问题是材料力学中的经典问题。线性理论能够给出欧拉临界载荷及屈曲模态,却不能确定失稳后的有限屈曲幅值。本文以两端铰支压杆为例,从势能出发,通过保留几何非线性产生的高阶项,说明欧拉临界点对应二阶势能项的退化,而四阶项则决定近临界后屈曲分支及其稳定性。由此,欧拉失稳可直观理解为势能由单势阱向双势阱的转变,并自然联系到Landau型势能和超临界叉形分岔。

     

    Abstract: Euler buckling is a classical problem in mechanics of materials. Linear theory can determine the Euler critical load and the corresponding buckling mode, but it cannot determine the finite post-buckling amplitude. Using a pinned–pinned rod as an example, we revisit this problem from an energy perspective. By retaining higher-order terms arising from geometric nonlinearity, we show that the Euler critical point corresponds to a degeneration of the quadratic term in the potential energy, while the quartic term determines the near-critical post-buckling branches and their stability. Euler buckling can therefore be understood intuitively as a transition of the energy landscape from a single well to a double well, naturally connecting the classical buckling problem to a Landau-type potential and a supercritical pitchfork bifurcation.

     

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